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Questions and Answers
Walking in the indicated direction has our left hand over the enclosed area.
Walking in the indicated direction has our left hand over the enclosed area.
True
Green’s Theorem cannot be used if the path has a negative orientation.
Green’s Theorem cannot be used if the path has a negative orientation.
True
The path described is oriented negatively.
The path described is oriented negatively.
False
The integral can be evaluated using Green's Theorem.
The integral can be evaluated using Green's Theorem.
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If the left hand is over the enclosed area, the orientation is negative.
If the left hand is over the enclosed area, the orientation is negative.
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Study Notes
Green's Theorem
- Green's theorem relates a line integral over a closed curve to a double integral over the region enclosed by the curve.
- The direction of a closed curve is positive if an observer walking along the curve with their head pointing in the direction of the positive normal to the surface has the surface on their left.
- Green's theorem states that the line integral of a vector field around a closed curve C is equal to the double integral of the curl of the vector field over the region enclosed by C.
- The formula for Green's theorem is: ∮ 𝐹⃗.𝑑𝑟⃗ = ∬𝑠 𝑐𝑢𝑟⃗𝑙 𝐹⃗.𝑛𝑑𝑠, where ∮ 𝐹⃗.𝑑𝑟⃗ is the line integral of the vector field 𝐹⃗ along the closed curve C, and ∬𝑠 𝑐𝑢𝑟⃗𝑙 𝐹⃗.𝑛𝑑𝑠 is the double integral of the curl of the vector field 𝐹⃗ over the region S enclosed by C.
- Unless otherwise stated, the line integral is assumed to be taken in the positive direction.
- Green's theorem is a special case of Stokes' theorem.
Gauss' Divergence Theorem
- Gauss' divergence theorem is a generalization of Green's theorem in the plane.
- Gauss' divergence theorem relates the flux of a vector field across a closed surface to the divergence of the vector field within the volume enclosed by the surface.
Relationship Between Green's Theorem and Gauss' Divergence Theorem
- Green's theorem in the plane deals with a two-dimensional region and its closed boundary curve.
- Gauss' divergence theorem deals with a three-dimensional region and its closed boundary surface.
- Green's theorem is essentially a special case of Gauss' divergence theorem where the region and its boundary are restricted to two dimensions.
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Description
This quiz focuses on Green's Theorem, a fundamental concept in vector calculus that connects line integrals over closed curves to double integrals over enclosed regions. It includes definitions, conditions for positive direction, and the theorem's formula. Additionally, the connection to Gauss' Divergence Theorem is mentioned.
